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<title>Concatenation of Maps</title>
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<p align="center"><font size="5" color="#0000FF"><b>DA</b></font><b><font color="#0000FF" size="5"> 
Concatenation: Constant Part Ignored</font></b></p>

<p>Consider the following two maps in one degree of freedom:</p>
<p><img border="0" src="mapmap.jpg" width="1080" height="332"></p>
<p>When using an automatic differentiation package such as FPP, one often 
computes the &quot;optical&quot; properties around a particular orbit, for example, the 
closed orbit of a circular ring.&nbsp; In such systems, the actual value of the 
zero<sup>th</sup> order part may be irrelevant. In fact, if we systematically 
ignore the constant part of polynomials in FPP operations, we do get a true 
&quot;Differential Algebra.&quot; Thus, without going into mathematical details, we refer 
to <b><font color="#CC0000">DA concatenation</font></b> any map concatenation in 
which <b><font color="#CC0000">the constant part of the map is ignored.</font></b></p>
<p>&nbsp;</p>

<p>Thus, ignoring the constant parts, the concatenation of M<sub>1</sub> and M<sub>2</sub>,&nbsp; 
M<sub>1</sub> o M<sub>2 </sub>is given by:</p>
<p align="center"><img border="0" src="m2m1.jpg" width="498" height="170"></p>
<p align="left">In FPP, we actually have:</p>
<p align="center"><img border="0" src="m2m1fpp.jpg" width="676" height="174"></p>
<p>The constant part of M<sub>2</sub> is added &quot;artificially&quot; at the end of the 
M<sub>1</sub> o M<sub>2 </sub>in FPP. The DA concatenation is always a <b>
<font color="#CC0000" size="4">&quot;*&quot;</font></b> in FPP. </p>
<p align="center"><font color="#0000FF" size="5"><b>
<a href="../simple_programs/z_damaps_da.f90">Program for this example</a></b></font></p>

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